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nLab > Latest Changes: microlocalization

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    • CommentRowNumber1.
    • CommentAuthorfpaugam
    • CommentTimeJul 31st 2011
    I have added a page on microlocalization a la Sato and Kashiwara-Schapira. It is complementary and different of the page on microlocal analysis (the approach is more algebraic). Perhaps both should be merged. Mathematically, the theory of ind-sheaves by Kashiwara-Schapira completes the bridge between classical analysis (a la Hormander) and Sato's approach, even if these two domains have quite different aims.

    I saw that there is also a page called algebraic microlocalization, but i don't like this name much (perhaps because of my ignorance): localization is allways algebraic (or sheaf theoretic), and microlocal analysis describes the corresponding thing in the analysis community.

    What do you think (in particular, Zoran)?
    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeAug 1st 2011

    Handy link: microlocalization.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeAug 2nd 2011
    • (edited Aug 2nd 2011)

    Microlocal analysis is a wider idea than just the procedure microlocalization. It deals with certain properties, which are microlocal. Algebraic microlocalization is discovered by Van den Bergh and van Oystaeyen much later than microlocalization of Kyoto school and which applies to broader algebraic context, which can not have the same interpretation as microlocality in analysis does and applies to certain noncommutative filtered rings.

    I do not understand what ind-sheaves have to do with microlocalization. They are useful indeed for the distribution spaces and so on, but this is not specific to microlocalization. The idea that ind-objects are close in spirit to bringing topologies into algebraic world and to extending spaces of distributions has many incarnations. For example, the de Rham cohomology for singular spaces can be modelled by taking the usual de Rham currents instead of forms, as shown by MacPherson and collaborators (for noncommutative exxtension of these ideas look at Cuntz-Quillen work from 1990s). Another case of this idea is the Morris-Pareigis formal scheme theory.

    • CommentRowNumber4.
    • CommentAuthorfpaugam
    • CommentTimeAug 12th 2011
    Ind-sheaves are useful in microlocalization theory a la Sato because they are necessary to define the functor mu, that is a quick way to define all microlocal invariants (e.g., microdifferential operators, microfunctions, etc...) in a coordinate free manner. Even to prove Riemann-Hilbert correspondence, one needs tempered, i.e., ind things. These microlocal invariants are in fact defined applying the functor mu to some ind-sheaves of tempered holomorphic functions. The point is that even for a usual sheaf, its microlocalization mu(F) is an ind-sheaf in general. This is not the same as mu_Z(F), previously defined by Sato.
    The other way to microlocalization a la Sato is by specialization and fourier transform. I prefer the Kashiwara-Schapira approach because it goes faster and allows the treatment of growth conditions, that are necessary for a coordinate free treatment of say, microdifferential operators (through cohomological methods).
    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeAug 12th 2011

    Thanks, put some of these remarks in nnLab…

    • CommentRowNumber6.
    • CommentAuthorfpaugam
    • CommentTimeAug 18th 2011
    > For example, the de Rham cohomology for singular spaces can be modelled by taking the usual de Rham currents instead of forms, > as shown by MacPherson and collaborators (for noncommutative exxtension of these ideas look at Cuntz-Quillen work from 1990s). > Another case of this idea is the Morris-Pareigis formal scheme theory.

    Can you be more specific on the relation between de Rham currents and ind-sheaves? And Cuntz-Quillen and ind-sheaves? Which work of Cuntz-Quillen are you talking about?

    Thanks for the reference to formal scheme theory!
    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeAug 18th 2011
    • J. Cuntz, D. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995), no. 2, 251–289, MR96c:19002, doi; Cyclic homology and nonsingularity, J. Amer. Math. Soc. 8 (1995), no. 2, 373–442, MR96e:19004, doi

    The second paper says:

    Our aim in this paper is to present the noncommutative analogue of the approach of Deligne [D] and Hartshorne [H] to de Rham cohomology in algebraic geometry. In this approach de Rham cohomology is first obtained for a nonsingular algebraic variety by means of the de Rham complex of differential forms. An arbitrary variety is then treated by embedding it in a nonsingular variety and completing the de Rham complex of the latter along the subvariety.

    On the other hand, another model, due MacPherson is explained in Chriss-Ginzburg book. One take just the de Rham complex of differential forms on a (complex) singular variety whose coefficients are generalized functions, so one works with the de Rham complex of currents. The first approach is having a spirit of ind-objects (and probably could be restated in their terms) and the second in the sense of distributions. Cuntz and Quillen notice that the noncommutative version of de Rham is the cyclic homology and try to do a similar program there.

    The Morris-Pareigis work generalized the Yoneda lemma to have some topology, so that they have a representing object a topological ring, not just a ring. It is like a formal power series ring representing some formal scheme. Further they represent formal schemes using formal colimit of certain presheaves, satisfying some axioms. It is well known that formal schemes can be considered as forming a subcategory of ind-schemes. The usual picture with formal spectra has some topologically ringed spaces, the topology is kind of replacing the flexibility which can be also achieved via formal systems. The connection is more explicit in a flexible approach of Morris-Pareigis.

    • CommentRowNumber8.
    • CommentAuthorfpaugam
    • CommentTimeAug 18th 2011
    Ok, i see, thanks!
    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeSep 20th 2022

    Related notions: algebraic microlocalization

    Page name changed from capitalized to ordinary as the usual convention in nnLab.

    diff, v10, current

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